Optical elements are pieces of substantially transparent material having surfaces that reflect or refract light, such as mirrors, lenses, splitters and collimators. Optical elements are used in a variety of applications, including telescopes, microscopes, cameras and spectacles.
Optical elements can be characterized by their optical properties and their surface optical properties. Optical properties such as astigmatism, optical power and prism describe how a wavefront of incident light is deformed as it passes through the optical element. Surface optical properties such as surface astigmatism, surface optical power and the gradients of the surface astigmatism and surface optical power describe geometrical properties of a surface of the optical element. The optical properties and the surface optical properties are closely related, but are not identical.
Multifocal optical elements have more than one optical power. For example, bifocal lenses have two subregions, each with a different optical power. Bifocal spectacle lenses can be used, for example, to correct myopia (short-sightedness) in one subregion, and presbyopia (the loss of the eye's ability to change the shape of its lens) in the other subregion. Unfortunately, many people find it uncomfortable to wear bifocal spectacle lenses, because of the abrupt change in optical power from one subregion to the other. This led to the development of progressive spectacle lenses, which are multifocal lenses in which the optical power varies smoothly from one point to another on the lens.
The surface optical power at any point on the surface of an optical element is defined by the mean curvature of the surface. A progressive lens has varying optical power, so it has variable curvature, and is by definition aspherical. However, since the surface of the progressive lens, or at least a substantial part of it, is by definition aspherical, it has two distinct principal curvatures .kappa..sub.1 and .kappa..sub.2 at many points. The surface astigmatism at any point on the surface of an optical element is defined by the absolute value of the difference in the principal curvatures .kappa..sub.1 and .kappa..sub.2.
The definitions of the mean curvature H, Gaussian curvature G, and principal curvatures .kappa..sub.1 and .kappa..sub.2 of a surface f at the point (x,y) are given in Equations 1A-1D: ##EQU1##
Since the early days of designing progressive lenses, the main design goals have been to achieve:
a) gently varying optical power; PA1 b) minimal astigmatism; PA1 c) reduction of a variety of optical aberrations such as skew distortion, binocular imbalance, etc.
Many different methods have been proposed to achieve these goals. U.S. Pat. No. 3,687,528 to Maitenaz describes a technique in which a base curve (meridian) runs from the upper part of the lens to its lower part. The lens surface is defined along the meridian such that the curvature varies gradually (and hence the optical power varies). Along the meridian itself, the principal curvatures .kappa..sub.1 and .kappa..sub.2 satisfy .kappa..sub.1 =.kappa..sub.2. The lens surface is extended from the meridian horizontally in several different methods. Explicit formulas are given for the extensions from the meridian. Maitenaz obtains an area in the upper part of the lens, and another area in the lower part of the lens in which there is a rather stable optical power. Furthermore, the astigmatism in the vicinity of the meridian is relatively small.
Many designs for progressive lenses explicitly divide the progressive lens into three zones: an upper zone for far vision, a lower zone for near vision, and an intermediate zone that bridges the first two zones. The upper and lower zones provide essentially clear vision. Many designs use spherical surfaces for the upper and lower zones. A major effort in the design process is to determine a good intermediate zone.
U.S. Pat. No. 4,315,673 to Guilino and U.S. Pat. No. 4,861,153 to Winthrop describe a method which achieves a smooth transition area through the use of explicit formula for the intermediate zone. U.S. Pat. No. 4,606,622 to Furter and G. Furter, "Zeiss Gradal HS--The progressive addition lens with maximum wearing comfort", Zeiss Information 97, 55-59, 1986, describe a method in which the lens designer defines the value of the lens surface in the intermediate zone at a number of special points. The full surface is then generated by the method of splines. The designer adjusts the value of the lens surface at the special points in order to improve the properties of the generated surface.
U.S. Pat. No. 4,838,675 to Barkan et al. describes yet another method. A progressive lens having an upper zone for far vision, a lower zone for near vision, and an intermediate zone is described by a base surface function. An improved progressive lens is calculated by optimizing a function defined over a subregion of the lens, where the optimized function is to be added to the base surface function.
A different technique is described by J. Loos, G. Greiner and H. P. Seidel, "A variational approach to progressive lens design", Computer Aided Design 30, 595-602, 1998 and by M. Tazeroualti, "Designing a progressive lens", in the book edited by P. J. Laurent et al., Curves and Surfaces in Geometric Design, AK Peters, 1994, pp. 467-474. The lens surface is defined as a combination of spline functions, and therefore the surface must be considered over a rectangle which is divided into smaller rectangles. This method is unnatural for those lenses which need to be defined over a shape other than a rectangle. A cost function is defined, and the spline coefficients are determined such that the surface minimizes the cost function. This method does not impose boundary conditions on the surface, and therefore lenses requiring a specific shape at the boundary cannot be designed using this method. Using cubic bisplines, this method provides an accuracy of h.sup.4, where h is the ratio of the diagonal of the smaller rectangle to the diagonal of the large rectangle.
European Patent Application EP744646 to Kaga et al. describes a method in which the surface of a progressive lens is partitioned into rectangles. At the boundaries of the rectangles, the surface must be continuous, differentiably continuous and twice differentiably continuous. Since optical power is related to curvature, and the curvatures are determined by the second derivatives of the surface, it seems natural to impose continuity constraints on the second derivatives. In fact, many method for designing progressive lens include continuity constraints on the second derivatives of the lens surface.